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Blowing-up solutions to competitive critical systems in dimension 3

Antonio J. Fernández, María Medina, Angela Pistoia

TL;DR

This work constructs new non-synchronized blowing-up solutions for competitive critical systems in dimension three via a Lyapunov–Schmidt reduction around a two-component bubble profile. The first component remains a Yamabe bubble, while the second aggregates $k$ bubbles concentrating at the vertices of a regular planar polygon; the remaining components arise by symmetry through rotations to ensure distinct profiles. A delicate balance of interactions yields a slow concentration rate $\tilde{\delta}_β^{1/2}|\log\tilde{\delta}_β|\sim-1/β$, and a finite-dimensional reduced energy $F_β(t)$ to locate true solutions; the method extends from $m=2$ to general $m\ge3$ by reducing to a two-component nonlocal system with additional symmetry. The results provide the first almost-explicit, non-synchronized blow-up constructions in 3D for competitive critical systems and illustrate how dimension critically shapes the reduction and interaction analysis.

Abstract

We study the critical system of $m\geq 2$ equations \begin{equation*} -Δu_i = u_i^5 + \sum_{j = 1,\,j\neq i}^m β_{ij} u_i^2 u_j^3\,, \quad u_i \gneqq 0 \quad \mbox{in } \mathbb{R}^3\,, \quad i \in \{1, \ldots, m\}\,, \end{equation*} where $β_{κ\ell} =α\in\mathbb{R}$ if $κ\neq\ell$, and $β_{\ell m}=β_{m κ} =β<0$, for $ κ, \ell \in \{1,\ldots, m-1\}$. We construct solutions to this system in the case where $β\to-\infty$ by means of a Ljapunov-Schmidt reduction argument. This allows us to identify the explicit form of the solution at main order: $u_1$ will look like a perturbation of the standard radial positive solution to the Yamabe equation, while $u_2$ will blow-up at the $k$ vertices of a regular planar polygon. The solutions to the other equations will replicate the blowing-up structure under an appropriate rotation that ensures $u_i\neq u_j$ for $i\neq j$. The result provides the first almost-explicit example of non-synchronized solutions to competitive critical systems in dimension 3.

Blowing-up solutions to competitive critical systems in dimension 3

TL;DR

This work constructs new non-synchronized blowing-up solutions for competitive critical systems in dimension three via a Lyapunov–Schmidt reduction around a two-component bubble profile. The first component remains a Yamabe bubble, while the second aggregates bubbles concentrating at the vertices of a regular planar polygon; the remaining components arise by symmetry through rotations to ensure distinct profiles. A delicate balance of interactions yields a slow concentration rate , and a finite-dimensional reduced energy to locate true solutions; the method extends from to general by reducing to a two-component nonlocal system with additional symmetry. The results provide the first almost-explicit, non-synchronized blow-up constructions in 3D for competitive critical systems and illustrate how dimension critically shapes the reduction and interaction analysis.

Abstract

We study the critical system of equations \begin{equation*} -Δu_i = u_i^5 + \sum_{j = 1,\,j\neq i}^m β_{ij} u_i^2 u_j^3\,, \quad u_i \gneqq 0 \quad \mbox{in } \mathbb{R}^3\,, \quad i \in \{1, \ldots, m\}\,, \end{equation*} where if , and , for . We construct solutions to this system in the case where by means of a Ljapunov-Schmidt reduction argument. This allows us to identify the explicit form of the solution at main order: will look like a perturbation of the standard radial positive solution to the Yamabe equation, while will blow-up at the vertices of a regular planar polygon. The solutions to the other equations will replicate the blowing-up structure under an appropriate rotation that ensures for . The result provides the first almost-explicit example of non-synchronized solutions to competitive critical systems in dimension 3.

Paper Structure

This paper contains 15 sections, 14 theorems, 222 equations.

Table of Contents

  1. Introduction
  2. Strategy of the proof
  3. Organization of the paper
  4. The case $m = 2$
  5. The error of approximation
  6. The linear theory
  7. Nonlinear theory
  8. Energy expansion
  9. Proof of Theorem \ref{['T.mainResult2eq']}
  10. The case $m\ge3$
  11. The error of approximation
  12. The linear theory
  13. Nonlinear theory
  14. Finite dimensional reduction
  15. Proof of Theorem \ref{['T.mainResultmgeq3']}

Key Result

Theorem 1.1

Assume that for certain $\alpha,\beta\in \mathbb R$ with $\beta<0$. Then, for any fixed integer $k \geqslant 2$, there exists $\beta_\star < -\sqrt{2}$ such that, for each $\beta \in (-\infty,\beta_\star]$, E.systemGeneralIntro has a finite energy solution $(u_{1,\beta},u_{2,\beta},\ldots,u_{m,\beta})$ of the f Here, $U$ is defined in E.bubbleIntro, ${\mathscr R}_{\frac{(i-2)}{m-1}\frac{2\pi}{k}}

Theorems & Definitions (28)

  • Theorem 1.1
  • Theorem 2.1: The case $m = 2$
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • Lemma 2.6
  • proof
  • ...and 18 more